Reed-Solomon Codes in Slow Frequency Hop Spread Spectrum Andrew Bolstad Iowa State University Advisor: Dr. John J. Komo Clemson University

Outline  Reed-Solomon Codes  System Description  Method of Calculation  Results and Conclusions

Reed-Solomon (RS) Codes  Error correcting  Structure of (n,k) Code Transmit n m-bit code symbols Represent k m-bit message symbols  Code Length “Natural” RS Code: n=2 m -1 Singly Extended RS Code: n=2 m Punctured/Shortened RS Code: n=2 m -2

Error Correction Capacity  e errors and r erasures, 2e+r≤n-k  Error – wrong symbol detected  Erasure – elimination of symbol  How can we choose symbols to erase? Parity bit: send n m+1-bit symbols Works for odd number of bit errors

Slow Frequency Hop Spread Spectrum  Many carrier frequencies (q), many users (K)  Transmit N b symbols, then “hop”  “Hit” Probability (Upper Bound): E. O. Geraniotis and M. B. Pursley, "Error Probabilities for Slow-Frequency- Hopped Spread-Spectrum Multiple-Access Communications Over Fading Channels" IEEE Trans. on Communication, vol. Com-30, no. 5, pp , May 1982.

System Block Diagram  Note: Interleave Transmission Symbols Reed Solomon encoder Transmission Symbol Generator Transmission Symbol Interleave SFH/SS Transmitter Original Message SFH/SS Receiver Transmission Symbol De-interleave Bit Stream Generator Reed Solomon decoder Received Message

Probability Calculations  Bits per code symbol divisible by bits per channel symbol s – probability of transmitting one symbol with no errors t – probability of erasing one symbol Easy formulas for P(not decoding)  Otherwise: analyze groups of 2 or 3

Results – Advantage of R-S Codes

Effects of MAI

Low MAI vs. High MAI

Conclusions  Reed Solomon Codes combat MAI  P(not dec.) limited by hit probability  QPSK / 8PSK better for high MAI

Future Work  Simulations to verify results  Equations / Algorithms for limits caused by MAI  Compare with bit / code symbol interleave  Determine optimal k for given n

The End

Transmission Schemes  M-ary Phase Shift Keying (MPSK)  Non-Coherent Scheme: DPSK  Bandwidth of MPSK

Probability of Error (AWGN)  BPSK/QPSK  DPSK  8PSK Bit Errors Per SymbolProbability 0P(A 0 ) 12P(A 1 )+P(A 3 ) 22P(A 2 )+P(A 4 ) 3P(A 3 ) A A A A A A A A 6 101